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For any $n>2$, let $X$ be an $n$-by-$n$ invertible matrix where every row has Euclidean norm $1$. Let $Y=X^{-1}$. Let $\Vert y_i \Vert$ be the Euclidean norm of column $i$ of $Y$.

The following conjecture seems to be confirmed by numerical evidence: for every $i$ $$ \frac{\Vert y_i \Vert}{\sum_{j=1}^n \Vert y_j \Vert} < \frac{1}{2}$$

How can we prove this?

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    With $n=1$ you get $1$, with $n=2$ it is always equal to $\frac12$ (not less). – 2017-02-08
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    Maybe trying to get an explicit relation between your matrix $X$ and $X^{-1}$ will let you verify what is going to be the Euclidean norm of every column of the inverse matrix, and by that, prove your inequality. It's just a hint... I don't know if it's going to work! – 2017-02-08

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